Ricci-flat metrics and dynamics on K3 surfaces
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Ricci-flat metrics and dynamics on K3 surfaces Valentino Tosatti1,2 Received: 16 March 2020 / Accepted: 10 October 2020 © Unione Matematica Italiana 2020
Abstract We give an overview of some recent interactions between the geometry of K 3 surfaces and their Ricci-flat Kähler metrics and the dynamical study of K 3 automorphisms with positive entropy.
1 Introduction K 3 surfaces form a distinguished class of compact complex surfaces which has received a tremendous amount of attention in several branches of mathematics. Our interest in K 3 surfaces stems from the fact that they are 2-dimensional Calabi–Yau manifold and hence admit Ricci-flat (but not flat) Kähler metrics, as we will explain below. The geometry of these metrics is still not completely understood, especially when families of such metrics degenerate. In a seemingly unrelated direction, K 3 surfaces have also been studied in holomorphic dynamics. The theory of holomorphic dynamics in 1 complex variable (on the Riemann sphere) is of course an enormous research area, and when one passes to 2 complex variables, it turns out that the only dynamically interesting automorphisms exist on K 3 and rational surfaces (see [10] for the precise statement), and interesting K 3 automorphism are relatively easy to construct. The dynamical study of such automorphisms was initiated by Cantat [11], and we refer the reader to the survey articles [12–14] and lecture notes [21] for a broader overview. The goal of this article will be to give an introduction to both of these aspects related to K 3 surfaces and to explain some recent work by Filip and the author [22,23] that exploits Ricci-flat metrics to prove results in dynamics and vice versa. In Sect. 2 we give an introduction to K 3 surfaces, including basic examples, the conjectures of Andreotti and Weil and their solutions. In Sect. 3 we discuss Yau’s Theorem [54] on the existence of Ricci-flat Kähler metrics on K 3 surfaces. Section 4 gives an overview of the dynamical study of automorphisms of K 3 surfaces, including basic properties and examples. Section 5 discusses the recent Kummer rigidity theorem of Cantat–Dupont [15] and Filip and the author [23], with an emphasis on the application of Ricci-flat metrics to this result that was found in [23]. In Sect. 6 we discuss applications of dynamics (in particular of Kummer
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Valentino Tosatti [email protected]
1
Department of Mathematics and Statistics, McGill University, Montréal, QC H3A 0B9, Canada
2
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA
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V. Tosatti
rigidity) to the problem of understanding the behavior of Ricci-flat Kähler metrics on K 3 surfaces when the Kähler class degenerates, following our work in [22]. Lastly, in Sect. 7 we discuss a few related open problems.
2 K 3 surfaces 2.1 Complex manifolds The main object of study in this article are K 3 surfaces (over the complex numbers). Before we get to the definition, let us briefly recall the basic definition of complex manifold. A comp
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